Hey guys how are you? I have the following question:(adsbygoogle = window.adsbygoogle || []).push({});

Let X1,X2,...,Xn be a random sample from a Pareto distribution having pdf

f(x|b)= (a*b^a)/x^(a+1) where x>=b (1)

Determine the maximum likelihood estimator for b, say b' on (0,infinity) and by considering P(b'>x) or otherwise show that b' has the Pareto distribution with pdf given by (1) but with a replaced by an.

My attempt: I found the MLE as b'=min Xi where 1<=i<=n, since our pdf is monotonically increasing w.r.t b.

After that I know how to find the asymptotic distribution of the MLE using the formula including the expected information but then we say that MLE follows a normal distribution for large n.

How do I show that the MLE follows a Pareto distribution in this case? I am so struggled, any help would be much appreciated!

P.S The hint tells us to consider P(b'>x) but how can I find P(min Xi >x) and why should it help me?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Distribution of Maximun Likelihood Estimator

**Physics Forums | Science Articles, Homework Help, Discussion**